| Preface Part Ⅰ: Compact Groups 1 Haar Measure 2 Schur Orthogonality 3 Compact Operators 4 The Peter-Weyl Theorem Part Ⅱ: Lie Group Fundamentals 5 Lie Subgroups of GL（n, C） 6 Vector Fields 7 Left-Invariant Vector Fields 8 The Exponential Map 9 Tensors and Universal Properties 10 The Universal Enveloping Algebra 11 Extension of Scalars 12 Representations of sl（2, C） 13 The Universal Cover 14 The Local Frobenius Theorem 15 Tori 16 Geodesics and Maximal Tori 17 Topological Proof of Cartan's Theorem 18 The Weyl Integration Formula 19 The Root System 20 Examples of Root Systems 21 Abstract Weyl Groups 22 The Fundamental Group 23 Semisimple Compact Groups 24 Highest-Weight Vectors 25 The Weyl Character Formula 26 Spin 27 Complexification 28 Coxeter Groups 29 The Iwasawa Decomposition 30 The Bruhat Decomposition 31 Symmetric Spaces 32 Relative Root Systems 33 Embeddings of Lie Groups Part Ⅲ: Topics 34 Mackey Theory 35 Characters of GL（n,C） 36 Duality between Sk and GL（n,C） 37 The Jacobi-Wrudi Identity 38 Schur Polynomials and GL（n,C） 39 Schur Polynomials and Sk 40 Random Matrix Theory 41 Minors of Toeplitz Matrices 42 Branching Formulae and Tableaux 43 The Cauchy Identity 44 Unitary Branching Rules 45 The Involution Model for Sk 46 Some Symmetric Algebras 47 Gelfand Pairs 48 Hecke Algebras 49 The Philosophy of Cusp Forms 50 Cohomology of Grassmannians References Index |
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